EXERCISES FOR TOPOLOGY III, WS05/06

Sheet 13, January 26 2006

Solutions due on Monday, 6th of February.

In this sheet we will do computations involving the classifying space BO(n) of the orthogonal group O(n). We recall (or admit) the following facts.

(a) O(1) ≅ ℤ/2, BO(1) ≅ ℝP^{∞}, and the inclusion O(1)^{n} → O(n) of diagonal matrices induces a continuous map d : (BO(1))^{n} → BO(n).

(b) The map d induces an injection of cohomology rings

d^{∗} : H^{∗}(BO(n); 𝔽_{2}) ↪ H^{∗}(BO(1)^{n}; 𝔽_{2}),

with image the subalgebra of H^{∗}(BO(1)^{n}; 𝔽_{2}) = P (u_{1}, …, u_{n}) generated by the symmetric polynomials σ_{1}, …, σ_{n}. These are defined by the equation

∏_{i=1}^{n}(1 + u_{i}) = ∑_{i=0}^{n} σ_{i}, with |σ_{i}| = i

(hence σ_{0} = 1, σ_{1} = u_{1} + ⋯ + u_{n}, …, σ_{n} = u_{1} … u_{n}).

Exercise 13.1. Prove that the subalgebra of P (u_{1}, …, u_{n}) generated by the symmetric polynomials σ_{1}, …, σ_{n} is free, so that

H^{∗}(BO(n); 𝔽_{2}) = P (w_{1}, w_{2}, w_{3}, …, w_{n})

with |w_{i}| = i and d^{∗}(w_{i}) = σ_{i} (the class w_{i} is called the i-th Stiefel-Whitney class).

Exercise 13.2. Prove the Wu-formula for the Stiefel-Whitney classes :

Sq^{j}w_{i} = ∑_{k=0}^{j} \binom{i − k − 1}{j − k}w_{i+j−k}w_{k} (0 ⩽ j ⩽ i).

Here by convention w_{0} = 1 and \binom{−1}{0} = 1.

Exercise 13.3. The space BO = ⋃_{n} BO(n) has an H-space structure induced by the direct sum of matrices. Indeed, the map O(m) × O(n) → O(m + n), (A, B) ↦

( \begin{matrix}A & 0 \\ 0 & B \end{matrix} ) induces a map BO(m) × BO(n) → BO(m + n). Taking the union

over m and n we obtain an H-space structure μ : BO × BO → BO. Show that H^{∗}(BO; 𝔽_{2}) = P (w_{1}, w_{2}, w_{3}, …), and that the coproduct ψ = μ^{∗} on H^{∗}(BO; 𝔽_{2}) is given by

ψ(w_{k}) = ∑_{i=0}^{k} w_{i} ⊗ w_{k−i}.

Exercise 13.4. Admitting that H_{∗}(BO; 𝔽_{2}) is of finite type, show that there is an isomorphism of algebras

H_{∗}(BO; 𝔽_{2}) ≅ P (x_{1}, x_{2}, x_{3} …)

(with the Pontrjagin product), where x_{i} is dual to w_{1}^{i} with respect to the basis of H^{∗}(BO; 𝔽_{2}) consisting of monomials in the Stiefel-Whitney classes.

Typeset by AMS-TEX